Abstract
This research proposes and delves into a stochastic competitive phytoplankton model with allelopathy and regime-switching. Sufficient criteria are proffered to ensure that the model possesses a unique ergodic stationary distribution (UESD). Furthermore, it is testified that these criteria are sharp on certain conditions. Some critical functions of regime-switching on the existence of a UESD of the model are disclosed: regime-switching could lead to the appearance of the UESD. The theoretical findings are also applied to research the evolution of Heterocapsa triquetra and Chrysocromulina polylepis.
Highlights
A global proliferation of harmful algal blooms (HABs) has caused significant harm to human and animal health, fisheries, tourism, ecosystem and environment in last decades [1]
Theorem 1 If b1 > 0 and b2 > 0, model (3) possesses a unique ergodic stationary distribution (UESD) concentrated on R2+ × X, where b1 = a1
We can deduce from Theorems 1 and 2 that (A) if b1 > 0 and b2 > 0, model (6) possesses a UESD concentrated on R2+ × X; (B) If b1 < 0 and b2 > 0, species 1 becomes extinct and the transition probability of
Summary
A global proliferation of harmful algal blooms (HABs) has caused significant harm to human and animal health, fisheries, tourism, ecosystem and environment in last decades [1]. Mandal and Banerjee [12] assumed that the environmental random disturbances are the white noise which mainly acts on the growth rates of the plankton with ri → ri + γiBi(t), they [12] tested the following stochastic competitive model with allelopathy:. Little research has been conducted to test the stationary distribution of model (2) or (3) For these reasons, in this paper, by taking advantage of some previous approaches and results mainly in [22], sufficient criteria are offered to ensure that model (3) possesses a unique ergodic stationary distribution (UESD) in Sect. We can deduce from Theorems 1 and 2 that (A) if b1 > 0 and b2 > 0, model (6) possesses a UESD concentrated on R2+ × X; (B) If b1 < 0 and b2 > 0, species 1 becomes extinct and the transition probability of ( 2(t), η(t)) converges weakly to 2;.
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